There are 18 points in a plane such that no three of them are in the same line except five points which are collinear. The number of triangles formed by these points is :
A.
816
B.
806
C.
805
D.
813
Answer :
806
Solution :
A triangle can be formed by using three non-collinear points.
So, the number of triangles formed by 18 non-collinear points $$ = {\,^{18}}{C_3}$$
But according to the question, 5 points are collinear.
Hence, exact number of triangles
$$\eqalign{
& = {\,^{18}}{C_3} - {\,^5}{C_3} = \frac{{18!}}{{3!15!}} - \frac{{5!}}{{3!2!}} \cr
& = \frac{{16 \times 17 \times 18}}{{2 \times 3}} - \frac{{4 \times 5}}{2} = 816 - 10 = 806. \cr} $$
Releted MCQ Question on Algebra >> Permutation and Combination
Releted Question 1
$$^n{C_{r - 1}} = 36,{\,^n}{C_r} = 84$$ and $$^n{C_{r + 1}} = 126,$$ then $$r$$ is:
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